A Brief New Proof to Fermat’s Last Theorem and Its Generalization

This article presents a brief and new solution to the problem known as the “Fermat’s Last Theorem”. It is achieved without the use of abstract algebra elements or elements from other fields of modern mathematics of the twentieth century. For this reason it can be easily understood by any mathematician or by anyone who knows basic mathematics. The important thing is that the above “theorem” is generalized. Thus, this generalization is essentially a new theorem in the field of number theory.


Introduction
Fermat's last theorem (known historically by this title) has been an unsolved puzzle in mathematics for over three centuries. The theorem itself is a deceptively simple formulation in mathematics, while Fermat famously stated that the problem had been solved around 1637. His claim was discovered 30 years after his death, as a clear statement on the margin of a book, but Fermat died without leaving any evidence as to his claim. This claim eventually became one of the most famous unsolved problems of mathematics. Efforts made to prove it, led to substantial development in number theory, and over time Fermat's Last Theorem gained legendary prominence as one of the most popular unsolved problems in mathematics [1]- [8].
Because this problem is easily understood by everyone (in terms of its word- We consider positive integers x, y, z that differ from each other and hypothesize that they verify the Equation (1.1) for a natural number 1 n > . Also, we hypothesize, without loss of the generality, that:

A Brief New Proof to Fermat's Last Theorem
converse is not always the case. For example, we consider that 3, 4, 5 x y z = = = and 3 n = . We have, 3  If we substitute z with y λ + or z y λ = + , where λ is a positive integer, then for the positive integers x, y, z, which according to the hypothesis we originally made, verify the Equation (1.1) for a natural number 1 n > , it is true that: So, given the above we have: We distinguish the following cases: Α. y n λ ≤ Considering Bernoulli's inequality is (for 1 n ≥ ): The first condition is rejected because x y = (not acceptable), from the second λ ≥ regardless from the exponent n. Then considering the conditions y n λ > and 2 y λ < − , because the number λ is greater or equal than number one for all n or 1 λ ≥ , 1 n ∀ > , we have: On the basis of inequalities 1 1 y n < and 1 1 3 y < , we distinguish the following conditions: 1 1 1 3 y n < < and 1 1 1 3 y n < ≤ . We observe that for 2 n = the first condition is satisfied while the second condition is not satisfied, on the contrary for 3 n ≥ , the second condition is satisfied while the first condition is not satisfied. Therefore, there is always at least one natural number n greater than the number one or 1 n > , so that, the condition 1 1 1 3 y n < < or 1 1 1 3 y n < ≤ is not satisfied. This means that the Equation (1.1) is not verified always for every natural number 1 n > and this is contrary to the sentence "when 1 λ ≥ regardless from the exponent n, the Equation (1.1) has solution for every natural number 1 n > " which arises from the hypothesis that

Proof of Theorem
We consider positive integers 1 2 3 1 , , , , , that differ from each other (m finite number) and hypothesize that they verify Equation (2.1) for a natural number 1 n > . Also, we hypothesize, without loss of the generality, that: Taking into account Equation (2.1) and the condition (2.2), on the basis of the above hypothesis we have: By combining conditions (2.3) and (2.4) we have: Comment: The double inequality (2.5) is sufficient but not necessary, i.e. the converse is not always the case. For example we consider that 1 3 x = , 2 4 x = , If we substitute m x with We distinguish the following cases: Α.
Chr. Poulkas Journal of Applied Mathematics and Physics Considering Bernoulli's inequality, it is (for 1 n ≥ ), By combining the conditions (2.7) and (2.8) we have, Because of condition (2.9), we observe that double inequality (2.5) is not satisfied, so in this case Equation (2.1) has no positive integer solutions 1 n ∀ > . Β.
obviously if they exist, this will be in case B, when the condition We will then prove that when positive integers 1 2 3 n > , the number λ is less than the differ- 1) First, we consider that 1 x λ > . Based on this condition, we have:    Given condition (2.19) we have: Based on condition (2.20) we distinguish the following sub cases:

1) From condition
We observe that solution of "Fermat's Last Theorem" occurs. This, to me, is a very strong indication that the solution of the generalization of Fermat's theorem is correct.
2) If ( 2 0 n m − + > or 2 1 n m − + ≥ ), is 1 n m ≥ − . In this case we have: 4) The immediately above example and the example which follows, namely: 6) In Equation (1.1), if 1 3 n < < , is 2 n = and so it has solutions that we known since ancient times as Pythagorean Triads. 7) Below, are presented some solutions of Equation (2.1), have made by prominent researchers, from time to time, of course using always the more times computer. It is easy to find that these solutions are perfectly in line with the general theorem.
Solutions of the Equation (

General Conclusion
The new solution to Fermat's Last Theorem, which presented here, is as brief and simple as its wording. It is achieved without the use of abstract algebra or elements from other fields of modern mathematics of the twentieth century. For this reason, it can be easily understood by any mathematician or by anyone who knows basic mathematics. This means that it has pedagogical value. At the same time, it is important, that the above "theorem" is generalized to an arbitrarily large number of variables. This generalization is essentially a new theorem in the field of the number theory, very useful to researchers of that field, because it gives answers to many open problems of the number theory. Also, it is important, that the solutions which were found by many prominent researchers in the past, are perfectly in line with the general theorem. -For m k = , we suppose that is true the following condition: